We consider the matrix equation X + A^* X^-n A = I, where I is the r×r unit matrix and A is an r×r invertible matrix. Several authors have studied the above matrix equation where n=1,  n=2 and they have obtained theoretical properties of these equtions. Nonlinear matrix equations of above type arise in many applications such as in control theory and statistics, in dynamic programming, stochastic filtering.

In this paper we derive two necessary and sufficient conditions for the existence of a positive definite solution. We prove two theorems which are necessary conditions for the considered equation.

We propose the following iterative methods

X0 = a I,     Xk+1 = I - A* Xk-n A ,    k=0, 1, ...

(1)

and

X0 = b I,     Xk+1 = ( A (I - Xk) A* )[ 1/n] ,    k=0, 1, ... .

(2)

There are matrices A for which the iterative procedure (1) converges to the positive definite matrix X^' and the iterative procedure (2) converges to the positive definite matrix X^''. We prove that X^' > X^'' (the matrix X^' - X^'' is positive definite).

We derive the sufficient conditions for the convergence rates of the algorithms where the methods are convergent.

Numerical examples are discussed and some results for the experiments are given.